Quantum Information Processing, Vol. 4, No. 5, November 2005 (© 2005)
Probabilities of Failure for Quantum Error Correction
A. J. Scott
Received June 13, 2005; accepted August 30, 2005
We investigate the performance of a quantum error-correcting code when pushed
beyond its intended capacity to protect information against errors, presenting for-
mulae for the probability of failure when the errors affect more qudits than that
speciﬁed by the code’s minimum distance. Such formulae provide a means to rank
different codes of the same minimum distance. We consider both error detection
and error correction, treating explicit examples in the case of stabilizer codes
constructed from qubits and encoding a single qubit.
KEY WORDS: Quantum error correction; quantum information.
Quantum error-correcting codes
protect quantum information against
noise. They play important roles in many areas of quantum information
theory, but most critically, in the viability of a quantum computer. Quan-
tum error correction negates a quantum state’s natural susceptibility to
decohere, and thus provides the long-time coherence necessary to sustain
quantum computation. Shor
presented the ﬁrst construc-
tions of quantum error-correcting codes. These discoveries led to a formal
connection between quantum codes and classical additive codes,
consequently, the characterization of a general class of quantum error-
correcting codes commonly referred to as stabilizer codes.
The idea behind quantum error correction is to encode quantum
states into qudits in such a way that a small number of errors affecting
the individual qudits can be detected and corrected to perfectly restore
the original encoded state. In this article we investigate the integrity of a
Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM
87131-1156, USA. E-mail: firstname.lastname@example.org
1570-0755/05/1100-0399/0 © 2005 Springer Science+Business Media, Inc.